Moment sequences and difference equations

TL;DR

This paper explores the relationship between moment sequences and difference equations, establishing conditions under which solutions preserve positivity and moment properties, with theoretical results supported by numerous examples.

Contribution

It provides a new criterion linking the roots of the characteristic equation to the support of the measure generating the moment sequence.

Findings

Roots of odd multiplicity must lie outside the measure's support

Homogeneous linear difference equations can generate positive moment sequences

Numerous examples illustrate the theoretical results

Abstract

We recall the definition and the properties of a moment sequence and recall that all real sequences that have a finite rank of its Hankel matrix (see definition in the sequel) satisfy a homogeneous linear equation with constant coefficients. Then we analyze cases when a difference equation with constant coefficients and suitably chosen initial conditions and having as an input a positive moment sequence has a solution that is a positive moment sequence. We give one general simple result and give many examples illustrating the theory. The main simple result states that the roots of the odd multiplicity of the characteristic equation must lie outside the support of the measure that produces the moment sequence that is in the input and the initial conditions suitably chosen.